Dynamical Properties of Error Statistics in a Shallow-Water Model

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  • 1 Department of Atmospheric Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, Los Angeles, California
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Abstract

Numerical ocean diagnoses and predictions rely on two types of information: model information and data information. Sequential estimation theory shows that the most probable state is a linear combination of the two, weighted according to their error statistics. A Kalman filter technique is applied to a one-layer reduced-gravity linear ocean model in a rectangular midlatitude basin. The model reproduces the main features of the subtropical wind-driven gyre; the filter is used to study the dynamical behavior of the error statistics.

On a midlatitude f plane, the error-correlation patterns among the state variables revealed by the Kalman filter are isotropic and homogeneous and satisfy a geostrophic relation. Introducing the β effect breaks the isotropy and homogeneity of the correlations, inducing behavior that is in agreement with two observational facts: 1) the latitudinal dependence of horizontal correlations and 2) the elliptic correlation shape of the mass field, elongated along the southwest–northeast orientation in the Northern Hemisphere. When a meridional line of observations is assimilated intermittently, the correlation patterns are dynamically adjusted to be wider to the east of the observing line than to the west. This is due to the westward propagation of errors by the model's Rossby wave dynamics.

The influence function of observations, based on the gain matrix of the Kalman filter, is subjected to polar decomposition into an amplitude part and a vector normalized by the amplitude—that is, a solid angle. The amplitude part contains the current observational information and determines the absolute weight given to an observation. The angular part is related to the previous observations only and reflects the structure of relative weights, whose behavior is similar to that of error correlations.

A criterion measuring the relative importance of different types of observations is defined, using Kalman filter techniques and geostrophic-error assumptions. The results from numerical experiments to examine the correctness of this criterion resolve apparent contradictions among the recent results of R. Daley, M. Ghil, and N. A. Phillips.

Abstract

Numerical ocean diagnoses and predictions rely on two types of information: model information and data information. Sequential estimation theory shows that the most probable state is a linear combination of the two, weighted according to their error statistics. A Kalman filter technique is applied to a one-layer reduced-gravity linear ocean model in a rectangular midlatitude basin. The model reproduces the main features of the subtropical wind-driven gyre; the filter is used to study the dynamical behavior of the error statistics.

On a midlatitude f plane, the error-correlation patterns among the state variables revealed by the Kalman filter are isotropic and homogeneous and satisfy a geostrophic relation. Introducing the β effect breaks the isotropy and homogeneity of the correlations, inducing behavior that is in agreement with two observational facts: 1) the latitudinal dependence of horizontal correlations and 2) the elliptic correlation shape of the mass field, elongated along the southwest–northeast orientation in the Northern Hemisphere. When a meridional line of observations is assimilated intermittently, the correlation patterns are dynamically adjusted to be wider to the east of the observing line than to the west. This is due to the westward propagation of errors by the model's Rossby wave dynamics.

The influence function of observations, based on the gain matrix of the Kalman filter, is subjected to polar decomposition into an amplitude part and a vector normalized by the amplitude—that is, a solid angle. The amplitude part contains the current observational information and determines the absolute weight given to an observation. The angular part is related to the previous observations only and reflects the structure of relative weights, whose behavior is similar to that of error correlations.

A criterion measuring the relative importance of different types of observations is defined, using Kalman filter techniques and geostrophic-error assumptions. The results from numerical experiments to examine the correctness of this criterion resolve apparent contradictions among the recent results of R. Daley, M. Ghil, and N. A. Phillips.

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